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Maurício Corrêa

Publications and source records attributed to Maurício Corrêa.

At least 19 recordsLinked to original sources
arXiv2026

Transverse analytic envelopes and integrability

We associate normal toric transverse analytic envelopes with the singular orbits of diagonal holomorphic actions, and construct their canonical flat transport. At independent singular supports, the semiglobal first-integral algebra is determined by periodic resonances; complete integrability is equivalent to full resonance rank and finite envelope monodromy. We classify labelled residual actions with a fixed envelope by Grassmannian families and reconstruct the ambient weights from the labelled quotient and logarithmic residues at full resonance rank. For commuting vector fields in convergent simultaneous semisimple normal form, we identify the algebra of holomorphic first integrals with the common kernel of the derivations induced on the toric envelope. Under compatible flat transport this yields a semiglobal criterion. Polynomial examples show that losses of first integrals may be invisible to any prescribed finite jet.

math.CV↗
arXiv2026

The topology of Kähler manifolds and 1-forms without zeros

We study how the topology and geometry of a compact Kähler manifold relate to the zeros of its one-forms. We determine all implications among several closely related conditions. In particular, we construct a smooth projective variety for which the Aomoto complex is exact for every nonzero holomorphic one-form and every semisimple local system, although every closed real one-form, and hence every holomorphic one-form, has a zero.

math.AG↗
arXiv2026

Invisible singularities in complex algebraic geometry

We construct morphisms between smooth complex projective varieties that have singular fibers, but look topologically smooth. We use this to give negative answers to the following four conjectures and questions: the smoothness conjecture of Fernández~de~Bobadilla and Kollár, a question of Kollár and Pardon on universal covers, Kotschick's conjecture on 1-forms, and a conjecture of Schreieder on Aomoto complexes. Version 2 is a revised version of sections 1--6 of the original, no change in the results. A substantially expanded version of sections 7--8 of the original will be posted separately.

math.AG↗
arXiv2026

On functoriality of Baum-Bott residues

We prove a functoriality formula for Baum--Bott residues under holomorphic pull-back. Let $f:V\to X$ be a holomorphic map generically transverse to a singular holomorphic foliation $\mathscr F$ on $X$. Under a compatibility condition on the tangent and normal sheaves, the localized Baum--Bott class of $f^*\mathscr F$ is the pull-back of that of $\mathscr F$, and the corresponding residue is obtained by localized intersection. We also discuss the universal relative class of Baum and Bott and its relation with the rationality conjecture. If the characteristic polynomial has rational coefficients, rational residues remain rational under the pull-backs considered here. For proper surjective generically finite maps of equal dimension, we obtain in addition a push-forward formula; hence rationality is equivalent before and after pull-back, in particular for bimeromorphic modifications satisfying the hypotheses of the theorem.

math.CV↗
arXiv2026

Geometry of the Donaldson-Friedman Pushout: Twistor degenerations and instanton charge

We study the Donaldson-Friedman semistable twistor degeneration by combining the scheme-theoretic geometry of its Ferrand pushout with the logarithmic topology of its Kato-Nakayama realisation. For the central fibre $ Z_0=\widetilde Z_1\cup_Q\widetilde Z_2,$ the Ferrand description yields an explicit equaliser presentation of the operational Chow ring and a componentwise specialisation formula, with sharp restrictions on surfaces that glue across the exceptional quadric. The logarithmic structure of the same normal-crossing fibre retains data not visible in ordinary intersection theory: after fixing the phase of the smoothing parameter, the Kato-Nakayama space over $Q$ is the unit circle bundle of the normal line bundle, and over a ruling fibre its anti-diagonal quotient is diffeomorphic to $\mathbb{RP}^3$. Restriction to curves in $Q$ gives a log-topological refinement of the algebraic intersection data. For bundles $E_0$ on $Z_0$ obtained by gluing Ward or Hartshorne-Serre data from the two components, we prove additivity of the second Chern cycle $c_2(E_0)\cap[Z_0]$. If $E_0$ extends over the semistable smoothing, its polarised charge is the sum of the component charges; under the usual reality and triviality conditions of the Ward correspondence, the bundle on a smooth fibre determines an anti-self-dual $SU(2)$-instanton on the connected sum of the underlying four-manifolds.

math.AG↗
arXiv2026

Log-Conformal Projective Manifolds

Let $(X,Δ)$ be a smooth complex projective simple normal crossing pair of dimension $n\ge 3$ endowed with an everywhere nondegenerate logarithmic conformal tensor. If $K_X+Δ$ is not nef, then exactly one of the following occurs: $Δ=\varnothing$ and $X\simeq Q^n$; $X\simeq\mathbb{P}^n$ and $Δ$ is a hyperplane; or $n=2m$ and $(X,Δ)$ admits a $(K_X+Δ)$-negative elementary contraction $ϕ:X\to Y$ generated by minimal rational curves of conformal degree one. In the third case, if $\dim Y>0$, then for some $1\le s\le m$ one has $\dim Y=m-s+1$ and the geometric generic fiber is $(\mathbb{P}^{m+s-1},H_1+\cdots+H_s)$, where the $H_i$ are hyperplanes in general position; over a dense open subset, $ϕ$ is a projective bundle and the horizontal boundary is the sum of $s$ relative hyperplanes. The case $s=1$ yields a rational maximal isotropic fibration with generic fiber $(\mathbb{P}^m,H)$; if $\dim Y=0$, then $ρ(X)=1$ and $-(K_X+Δ)$ is ample. If $K_X+Δ\equiv0$, then, under a Bochner extension principle, a restricted-holonomy condition, and trivial monodromy of the induced flat connection on $X\setminusΔ$, a logarithmic conformal tensor with trivial conformal line bundle forces $X\setminusΔ$ to be semi-abelian and $(X,Δ)$ to be its toroidal compactification.

math.AG↗
arXiv2026

Moduli, Deformations and Algebraic K-Theory of Graded PI-Algebras

Over a field of characteristic zero, for a finite grading group and a finitely generated graded $T$-ideal, we construct relatively free sheaves and finite-rank quotient stacks of algebra laws with prescribed identities. Their tangent complexes are PI-restricted Hochschild complexes, and the second variations of the identities give explicit lifting obstructions. The first-order deformation groupoid carries a natural relative algebraic K-theory functor; for its square-zero extensions, the relative Chern character identifies rational relative K-theory with negative cyclic homology. Additional identities determine closed PI-strata and localisation fibre sequences in K-theory with Azumaya coefficients. On the Azumaya substack, the degree annihilates the Brauer class, and Morita transport yields a base-change-compatible K-theory of PI-Azumaya families together with a K-theoretic monodromy equivalence.

math.AG↗
arXiv2026

A lower bound for the Milnor number of vector fields

We study holomorphic vector fields whose singular locus contains a smooth positive-dimensional local complete-intersection component. We obtain an exact global formula for its Milnor-Poincaré-Hopf contribution and prove a sharp local lower bound, valid under holomorphic perturbations, in terms of the embedded contribution associated with the component. As an application to gradient vector fields, we derive a lower bound for the total Milnor number of the isolated critical points arising near a smooth positive-dimensional critical component; for a morsification, this becomes a lower bound for the number of Morse critical points converging to that component. Explicit families show that the bounds are sharp and exhibit the redistribution of singularities between a fixed neighbourhood of the component and the hyperplane at infinity in projective compactifications.

math.AG↗
arXiv2026

Foliated and Mather-Jacobian discrepancies via tangential arcs

This article develops a tangential arc-space approach to foliated discrepancies for logarithmic simple co-rank one foliations on threefolds, relative to a fixed invariant normal crossing separatrix divisor. In the non-resonant logarithmic case, reduced tangential arcs centred on the prescribed tangential locus are confined to this divisor. The tangential sector is therefore presented, at the reduced arc level, by the normalised separatrix-conductor system. Foliated adjunction transfers the discrepancy calculus to ordinary log pairs obtained by adjunction on the normalised branches and conductors. The arc-space theorem of Ein-Mustaţă-Yasuda, applied on these strata, then gives a tangential codimension formula identifying logarithmic codimensions of toroidal tangential divisorial cylinders with the corresponding tangential discrepancies. For toroidal invariant divisors read on branches, this tangential discrepancy agrees with the usual foliated discrepancy. The resulting theory gives toroidal tangential inversion of adjunction, a branch--conductor description of the tangential non-lc and non-klt loci, a cylinder criterion for tangential log canonicity, lower semicontinuity of the toroidal tangential minimal log discrepancy, and a relative Mather-Jacobian refinement for the canonical image separatrix system.

math.AG↗
arXiv2026

On residue localization for complex supermanifolds

We prove a holomorphic residue localization formula for odd holomorphic vector fields on compact complex supermanifolds whose fermionic and bosonic dimensions coincide. Under isolated non-degeneracy hypotheses on the reduced zero set, we give an explicit local residue formula.

math.DG↗
arXiv2026

Transversal unfoldings of derived foliations

We establish a homotopy-invariant theory of transversal unfoldings of relative derived foliations. A transversal unfolding is an integrable lift of the parameter directions to infinitesimal transverse symmetries of the relative foliation. We construct a canonical transverse controller and prove that the groupoid of strict transversal unfoldings is the one-truncation of its derived space of flat splittings. For affine Chevalley--Eilenberg presentations, the controller is described by first-order Lie derivations modulo inner derivations. When the inner action has a kernel, the ordinary quotient is replaced by a crossed controller whose derived mapping space retains the transported central isotropy and the ensuing higher homotopies. We then construct the controller intrinsically from the graded-mixed de Rham algebra, prove its independence of the chosen presentation, and obtain a global classification under explicit descent hypotheses. The theory encompasses the classical, logarithmic, shifted-Poisson, and representable leaf-space cases.

math.AG↗
arXiv2026

A Poisson--Poincaré--Dulac for Poisson Connections

We study Poisson-flat connections with logarithmic poles along a simple normal crossings divisor on a holomorphic Poisson manifold, where flatness is required only along the symplectic foliation. After identifying the relevant logarithmic cotangent Poisson Lie algebroid, we define an Euler--Poisson principal part and a residue theory adapted to the canonical logarithmic Hamiltonian generators. Under a precise nonresonance hypothesis, we establish a Poisson Poincaré--Dulac theorem: any logarithmic Poisson-flat connection with prescribed principal part is holomorphically gauge equivalent to a pure Euler--Poisson normal form with constant commuting residues, and this normal form is unique up to Casimir-valued gauge transformations lying in the common centralizer of the residues. To encode both leafwise transport and boundary winding, we construct a twisted leafwise fundamental groupoid via the real oriented blow-up and a 2-pushout that adjoins canonical tangential meridians. In the nonresonant regime, the normal form yields a meridional character determined by the residues, and hence a logarithmic Riemann--Hilbert correspondence in the Poisson setting for this groupoid. Finally, we illustrate the theory with rank-two Poisson modules (Poisson triples) and an explicit family of examples.

math.AG↗
arXiv2026

Stacky geometry and logarithmic topology of transversely affine foliations

We study the stacky and logarithmic-topological structures associated with transversely affine affine foliations. To such a foliation we attach its holonomy group and the corresponding quotient stack, which provides the natural geometric base for the multiplicative developing coordinate. Holomorphic and meromorphic reparametrisations are then identified with endomorphisms of this stack and of its compactification, yielding a geometric Singer-type theorem. We then classify these reparametrisations according to the geometry of the holonomy group. On the logarithmic-topological side, we pass to the Kato-Nakayama space, where the residues define canonical boundary characters, govern the induced linear dynamics on the boundary tori, and give rise to a canonical logarithmic lift of the developing map. In this way, the quotient stack controls the linear part of the theory, while the Kato-Nakayama space captures its logarithmic-topological and dynamical content.

math.AG↗
arXiv2026

Excess logarithmic residues for foliations by curves and applications

We introduce excess logarithmic residues for one-dimensional holomorphic foliations tangent to a divisor. They arise from the comparison between the logarithmic normal sheaf and the ordinary normal sheaf of the foliation, and measure the local variation between the logarithmic and classical Baum--Bott contributions. We prove a global residue formula expressing the corresponding Chern numbers as sums of local residues. We then derive a Poincaré-type bound for invariant hypersurfaces from the non-negativity of the relevant logarithmic residues. Finally, for a normal \(\mathbb Q\)-Gorenstein surface $Y$, we show that the componentwise logarithmic residues of a lifted foliation along the exceptional divisor of a functorial resolution recover the log discrepancies of the singularities of $Y$, giving a dynamical and foliated test for log canonicity of these singularities.

math.AG↗
arXiv2026

On the connectedness of the singular set of holomorphic foliations

Let $\mathcal{F}$ be a singular holomorphic foliation of dimension $k>1$ on a projective $n$-manifold $X$. Assume that the determinant of the normal sheaf of $\mathcal{F}$ is ample (as is always the case when $X=\mathbb{P}^{n}$), and that the singular set $Sing(\mathcal{F})$ has dimension $\leq k-1$. We show that the union of those irreducible components of $Sing(\mathcal{F})$ of dimension exactly $k-1$ is necessarily connected. Consequently, we obtain a Bott-type topological obstruction to the integrability of singular holomorphic distributions, echoing Bott's vanishing theorem, and we answer a question of Cerveau for codimension-one foliations on $\mathbb{P}^{3}$.

math.AG↗
arXiv2026

Log Bott localization with non-isolated lci zero varieties

We establish a logarithmic Bott localization formula for global holomorphic sections of $T_X(-\log D)$ on a compact complex manifold $X$ with simple normal crossings divisor $D$. The zero scheme is allowed to have non-isolated compact components, assumed to be local complete intersections and to satisfy the natural Bott nondegeneracy condition. We further give a current-theoretic formulation and, in the local complete intersection case, identify the local residue term with a Coleff-Herrera current.

math.CV↗
arXiv2026

Log Baum--Bott Residues for foliations by curves

We prove a Baum--Bott type residual formula for one-dimensional holomorphic foliations, and logarithmic along free divisors. More precisely, this provides a Baum--Bott theorem for a foliated triple $(X, \mathcal{F}, D)$, where $\mathcal{F}$ is a foliation by curves and $D$ is a free divisor on a complex manifold $X$. From the local point of view, we show that the log Baum--Bott residues are a generalization of the Aleksandrov logarithmic index for vector fields with isolated singularities on hypersurfaces. We also show how these new indices are related to Poincaré's Problem for foliations by curves. In the case of foliated surfaces, we show that the differences between the logarithmic residues and Baum--Bott indices along invariant curves can be expressed in terms of the GSV and Camacho--Sad indices. We also obtain a Baum--Bott type formula for singular varieties via log resolutions. Finally, we prove a weak global version of the Zariski--Lipman conjecture for compact algebraic surfaces, in the form of a foliated smoothness criterion, suggesting the appearance of saddle-nodes in the singularity reduction on singular surfaces.

math.AG↗